Let a,b,c be real numbers greater than 1. Let S=log_a bc+log_b ac+log_c ba Then find the minimum value of S

cortejosni

cortejosni

Answered question

2022-08-13

Find the minimum value of S
Let a , b , c be real numbers greater than 1.
Let S = log a b c + log b a c + log c b a
Then find the minimum value of S

Answer & Explanation

beentjie8e

beentjie8e

Beginner2022-08-14Added 20 answers

First, we divide the logs of the multiplications into sums:
log a b c + log b a c + log c b a = log a b + log a c + log b a + l o g b c + log c a + log c b
After that, we use the l o g a b = ln a ln b to further deconstruction:
log a b + log a c + log b a + l o g b c + log c a + log c b = ln b ln a + ln c ln a + ln a ln b + ln c ln b + ln a ln c + ln b ln c
Because a, b and c are reals >1, their natural logarithm is a real > 0.
Substituting x := ln a ln b , y := ln a ln c , z := ln b ln c , we are looking for the minimum of x + 1 x + y + 1 y + z + 1 z
The minimum of a number plus its recipe is 2 for positive real numbers [prove is trivial, I give you on need], thus the minimum is 6 _ _
P.s. also trivial that we get this minimum if a = b = c

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